= Asymptotic mean integrated squared error
{title2=$\operatorname{AMISE}(h)=R(K)/(nh)+\mu_2(K)^2R(f'')h^4/4$}
= AMISE
{c}
{synonym}
For a second-order <kernel density estimator>, write $R(q)=\int q^2$ and $\mu_2(K)=\int u^2K(u)\,du$. Under the usual smoothness and shrinking-bandwidth assumptions, the leading <mean integrated squared error> is the displayed sum of integrated <variance> and squared <bias of an estimator>. Its minimizer is $h=[R(K)/(n\mu_2(K)^2R(f''))]^{1/5}$, giving order $n^{-4/5}$. An added $O(n^{-1})$ integrated covariance term is lower order at this bandwidth and does not change its leading constant.
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